3.37 \(\int (a \text {csch}^3(x))^{3/2} \, dx\)

Optimal. Leaf size=81 \[ \frac {10}{21} a \cosh (x) \sqrt {a \text {csch}^3(x)}-\frac {2}{7} a \coth (x) \text {csch}(x) \sqrt {a \text {csch}^3(x)}+\frac {10}{21} i a \sqrt {i \sinh (x)} \sinh (x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \text {csch}^3(x)} \]

[Out]

10/21*a*cosh(x)*(a*csch(x)^3)^(1/2)-2/7*a*coth(x)*csch(x)*(a*csch(x)^3)^(1/2)+10/21*I*a*(sin(1/4*Pi+1/2*I*x)^2
)^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(cos(1/4*Pi+1/2*I*x),2^(1/2))*sinh(x)*(a*csch(x)^3)^(1/2)*(I*sinh(x))^(1/
2)

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Rubi [A]  time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3768, 3771, 2641} \[ \frac {10}{21} a \cosh (x) \sqrt {a \text {csch}^3(x)}-\frac {2}{7} a \coth (x) \text {csch}(x) \sqrt {a \text {csch}^3(x)}+\frac {10}{21} i a \sqrt {i \sinh (x)} \sinh (x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \text {csch}^3(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csch[x]^3)^(3/2),x]

[Out]

(10*a*Cosh[x]*Sqrt[a*Csch[x]^3])/21 - (2*a*Coth[x]*Csch[x]*Sqrt[a*Csch[x]^3])/7 + ((10*I)/21)*a*Sqrt[a*Csch[x]
^3]*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]]*Sinh[x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \left (a \text {csch}^3(x)\right )^{3/2} \, dx &=\frac {\left (i a \sqrt {a \text {csch}^3(x)}\right ) \int (i \text {csch}(x))^{9/2} \, dx}{(i \text {csch}(x))^{3/2}}\\ &=-\frac {2}{7} a \coth (x) \text {csch}(x) \sqrt {a \text {csch}^3(x)}+\frac {\left (5 i a \sqrt {a \text {csch}^3(x)}\right ) \int (i \text {csch}(x))^{5/2} \, dx}{7 (i \text {csch}(x))^{3/2}}\\ &=\frac {10}{21} a \cosh (x) \sqrt {a \text {csch}^3(x)}-\frac {2}{7} a \coth (x) \text {csch}(x) \sqrt {a \text {csch}^3(x)}+\frac {\left (5 i a \sqrt {a \text {csch}^3(x)}\right ) \int \sqrt {i \text {csch}(x)} \, dx}{21 (i \text {csch}(x))^{3/2}}\\ &=\frac {10}{21} a \cosh (x) \sqrt {a \text {csch}^3(x)}-\frac {2}{7} a \coth (x) \text {csch}(x) \sqrt {a \text {csch}^3(x)}+\frac {1}{21} \left (5 a \sqrt {a \text {csch}^3(x)} \sqrt {i \sinh (x)} \sinh (x)\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx\\ &=\frac {10}{21} a \cosh (x) \sqrt {a \text {csch}^3(x)}-\frac {2}{7} a \coth (x) \text {csch}(x) \sqrt {a \text {csch}^3(x)}+\frac {10}{21} i a \sqrt {a \text {csch}^3(x)} F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {i \sinh (x)} \sinh (x)\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 56, normalized size = 0.69 \[ -\frac {2}{21} a \sinh (x) \sqrt {a \text {csch}^3(x)} \left (\coth (x) \left (3 \text {csch}^2(x)-5\right )-5 i \sqrt {i \sinh (x)} F\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csch[x]^3)^(3/2),x]

[Out]

(-2*a*Sqrt[a*Csch[x]^3]*(Coth[x]*(-5 + 3*Csch[x]^2) - (5*I)*EllipticF[(Pi - (2*I)*x)/4, 2]*Sqrt[I*Sinh[x]])*Si
nh[x])/21

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \operatorname {csch}\relax (x)^{3}} a \operatorname {csch}\relax (x)^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^3)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csch(x)^3)*a*csch(x)^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {csch}\relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^3)^(3/2),x, algorithm="giac")

[Out]

integrate((a*csch(x)^3)^(3/2), x)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (a \mathrm {csch}\relax (x )^{3}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csch(x)^3)^(3/2),x)

[Out]

int((a*csch(x)^3)^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {csch}\relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^3)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*csch(x)^3)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {a}{{\mathrm {sinh}\relax (x)}^3}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/sinh(x)^3)^(3/2),x)

[Out]

int((a/sinh(x)^3)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {csch}^{3}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)**3)**(3/2),x)

[Out]

Integral((a*csch(x)**3)**(3/2), x)

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